The Golden Ratio and the Golden Ring of Russia In the book of Erich von Däniken (see) I read that sacred places in Ancient Greece are connected to each other by the proportion of the Golden Ratio. I quote the personally verified data that is given in this book (see Fig. 55 and 56): 1. Line

This harmony is striking in its scale...

Hello, friends!

Have you heard anything about Divine Harmony or the Golden Ratio? Have you ever thought about why something seems ideal and beautiful to us, but something repels us?

If not, then you have successfully come to this article, because in it we will discuss the golden ratio, find out what it is, what it looks like in nature and in humans. Let's talk about its principles, find out what the Fibonacci series is and much more, including the concept of the golden rectangle and the golden spiral.

Yes, the article has a lot of images, formulas, after all, the golden ratio is also mathematics. But everything is described in fairly simple language, clearly. And at the end of the article, you will find out why everyone loves cats so much =)

What is the golden ratio?

To put it simply, the golden ratio is a certain rule of proportion that creates harmony?. That is, if we do not violate the rules of these proportions, then we get a very harmonious composition.

The most comprehensive definition of the golden ratio states that the smaller part is related to the larger one, as the larger part is to the whole.

But besides this, the golden ratio is mathematics: it has a specific formula and a specific number. Many mathematicians, in general, consider it the formula of divine harmony, and call it “asymmetrical symmetry”.

The golden ratio has reached our contemporaries since the times of Ancient Greece, however, there is an opinion that the Greeks themselves had already spied the golden ratio among the Egyptians. Because many works of art Ancient Egypt clearly constructed according to the canons of this proportion.

It is believed that Pythagoras was the first to introduce the concept of the golden ratio. The works of Euclid have survived to this day (he used the golden ratio to build regular pentagons, which is why such a pentagon is called “golden”), and the number of the golden ratio is named after the ancient Greek architect Phidias. That is, this is our number “phi” (denoted Greek letterφ), and it is equal to 1.6180339887498948482... Naturally, this value is rounded: φ = 1.618 or φ = 1.62, and in percentage terms the golden ratio looks like 62% and 38%.

What is unique about this proportion (and believe me, it exists)? Let's first try to figure it out using an example of a segment. So, we take a segment and divide it into unequal parts in such a way that its smaller part relates to the larger one, as the larger part relates to the whole. I understand, it’s not very clear yet what’s what, I’ll try to illustrate it more clearly using the example of segments:

So, we take a segment and divide it into two others, so that the smaller segment a relates to the larger segment b, just as the segment b relates to the whole, that is, the entire line (a + b). Mathematically it looks like this:

This rule works indefinitely; you can divide segments as long as you like. And, see how simple it is. The main thing is to understand once and that’s it.

But now let's take a closer look complex example, which comes across very often, since the golden ratio is also represented in the form of a golden rectangle (the aspect ratio of which is φ = 1.62). This is a very interesting rectangle: if we “cut off” a square from it, we will again get a golden rectangle. And so on endlessly. See:

But mathematics would not be mathematics if it did not have formulas. So, friends, now it will “hurt” a little. I hid the solution to the golden ratio under a spoiler; there are a lot of formulas, but I don’t want to leave the article without them.

Fibonacci series and golden ratio

We continue to create and observe the magic of mathematics and the golden ratio. In the Middle Ages there was such a comrade - Fibonacci (or Fibonacci, they spell it differently everywhere). He loved mathematics and problems, he also had an interesting problem with the reproduction of rabbits =) But that’s not the point. He opened number sequence, the numbers in it are called “Fibonacci numbers”.

The sequence itself looks like this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233... and so on ad infinitum.

In other words, the Fibonacci sequence is a sequence of numbers where each subsequent number is equal to the sum of the previous two.

What does the golden ratio have to do with it? You'll see now.

Fibonacci Spiral

To see and feel the whole connection between the Fibonacci number series and the golden ratio, you need to look at the formulas again.

In other words, from the 9th term of the Fibonacci sequence we begin to obtain the values ​​of the golden ratio. And if we visualize this whole picture, we will see how the Fibonacci sequence creates rectangles closer and closer to the golden rectangle. This is the connection.

Now let's talk about the Fibonacci spiral, it is also called the “golden spiral”.

The golden spiral is a logarithmic spiral whose growth coefficient is φ4, where φ is the golden ratio.

In general, from a mathematical point of view, the golden ratio is an ideal proportion. But this is just the beginning of her miracles. Almost the entire world is subject to the principles of the golden ratio; nature itself created this proportion. Even esotericists see numerical power in it. But we will definitely not talk about this in this article, so in order not to miss anything, you can subscribe to site updates.

Golden ratio in nature, man, art

Before we begin, I would like to clarify a number of inaccuracies. Firstly, the very definition of the golden ratio in this context is not entirely correct. The fact is that the very concept of “section” is a geometric term, always denoting a plane, but not a sequence of Fibonacci numbers.

And secondly, number series and the ratio of one to the other, of course, has been turned into a kind of stencil that can be applied to everything that seems suspicious, and one can be very happy when there are coincidences, but still, common sense should not be lost.

However, “everything was mixed up in our kingdom” and one became synonymous with the other. So, in general, the meaning is not lost from this. Now let's get down to business.

You will be surprised, but the golden ratio, or rather the proportions as close as possible to it, can be seen almost everywhere, even in the mirror. Don't believe me? Let's start with this.

You know, when I was learning to draw, they explained to us how easier it is to build a person’s face, his body, and so on. Everything must be calculated relative to something else.

Everything, absolutely everything is proportional: bones, our fingers, palms, distances on the face, the distance of outstretched arms in relation to the body, and so on. But even that's not all internal structure of our body, even it, is equal or almost equal to the golden section formula. Here are the distances and proportions:

from shoulders to crown to head size = 1:1.618

from the navel to the crown to the segment from the shoulders to the crown = 1:1.618

from navel to knees and from knees to feet = 1:1.618

from chin to extreme point upper lip and from it to the nose = 1:1.618

Isn't this amazing!? Harmony in its purest form, both inside and outside. And that is why, at some subconscious level, some people do not seem beautiful to us, even if they have a strong, toned body, velvety skin, beautiful hair, eyes, etc., and everything else. But, all the same, the slightest violation of the proportions of the body, and the appearance already slightly “hurts the eyes.”

In short, the more beautiful a person seems to us, the closer his proportions are to ideal. And this, by the way, can be attributed not only to the human body.

Golden ratio in nature and its phenomena

A classic example of the golden ratio in nature is the shell of the mollusk Nautilus pompilius and the ammonite. But this is not all, there are many more examples:

in the curls of the human ear we can see a golden spiral;

its same (or close to it) in the spirals along which galaxies twist;

and in the DNA molecule;

According to the Fibonacci series, the center of a sunflower is arranged, cones grow, the middle of flowers, a pineapple and many other fruits.

Friends, there are so many examples that I’ll just leave the video here (it’s just below) so as not to overload the article with text. Because if you dig into this topic, you can go deeper into the following jungle: even the ancient Greeks proved that the Universe and, in general, all space is planned according to the principle of the golden ratio.

You will be surprised, but these rules can be found even in sound. See:

The highest point of sound that causes pain and discomfort in our ears is 130 decibels.

We divide the proportion 130 by the golden ratio number φ = 1.62 and we get 80 decibels - the sound of a human scream.

We continue to divide proportionally and get, let’s say, the normal volume of human speech: 80 / φ = 50 decibels.

Well, the last sound that we get thanks to the formula is a pleasant whispering sound = 2.618.

Using this principle, it is possible to determine the optimal-comfortable, minimum and maximum numbers of temperature, pressure, and humidity. I haven’t tested it, and I don’t know how true this theory is, but you must agree, it sounds impressive.

One can read the highest beauty and harmony in absolutely everything living and non-living.

The main thing is not to get carried away with this, because if we want to see something in something, we will see it, even if it is not there. For example, I paid attention to the design of the PS4 and saw the golden ratio there =) However, this console is so cool that I wouldn’t be surprised if the designer really did something clever there.

Golden ratio in art

This is also a very large and extensive topic that is worth considering separately. Here I will just note a few basic points. The most remarkable thing is that many works of art and architectural masterpieces of antiquity (and not only) were made according to the principles of the golden ratio.

Egyptian and Mayan pyramids, Notre Dame de Paris, Greek Parthenon and so on.

In the musical works of Mozart, Chopin, Schubert, Bach and others.

In painting (this is clearly visible): all the most famous paintings by famous artists are made taking into account the rules of the golden ratio.

These principles can be found in Pushkin’s poems and in the bust of the beautiful Nefertiti.

Even now, the rules of the golden ratio are used, for example, in photography. Well, and of course, in all other arts, including cinematography and design.

Golden Fibonacci cats

And finally, about cats! Have you ever wondered why everyone loves cats so much? They've taken over the Internet! Cats are everywhere and it's wonderful =)

And the whole point is that cats are perfect! Don't believe me? Now I’ll prove it to you mathematically!

Do you see? The secret is revealed! Cats are ideal from the point of view of mathematics, nature and the Universe =)

*I'm kidding, of course. No, cats are really ideal) But no one has measured them mathematically, probably.

That's basically it, friends! We'll see you in the next articles. Good luck to you!

P.S. Images taken from medium.com.

Recently, a close friend decided to get herself a tattoo of the “golden ratio” and I, becoming interested, decided to educate myself on this subject. Here is a little material about the history of the issue.

The golden ratio is a universal manifestation of structural harmony. It is found in nature, science, art - in everything that a person can come into contact with. Once having become acquainted with the golden rule, humanity no longer betrayed it.
The golden ratio is a universal manifestation of structural harmony. It is found in nature, science, art - in everything that a person can come into contact with. Once having become acquainted with the golden rule, humanity no longer betrayed it.
Definition
The most comprehensive definition of the golden ratio states that the smaller part is related to the larger one, as the larger part is to the whole. Its approximate value is 1.6180339887. In a rounded percentage value, the proportions of the parts of the whole will correspond as 62% to 38%. This relationship operates in the forms of space and time.
The ancients saw the golden ratio as a reflection of cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science considers the golden ratio as “asymmetrical symmetry”, calling it in a broad sense a universal rule reflecting the structure and order of our world order.
Story
The ancient Egyptians had an idea about the golden proportions, they knew about them in Rus', but for the first time the golden ratio was scientifically explained by the monk Luca Pacioli in the book “Divine Proportion” (1509), illustrations for which were supposedly made by Leonardo da Vinci. Pacioli saw in the golden section the divine trinity: the small segment personified the Son, the large segment the Father, and the whole the Holy Spirit.
The name of the Italian mathematician Leonardo Fibonacci is directly associated with the golden ratio rule. As a result of solving one of the problems, the scientist came up with a sequence of numbers now known as the Fibonacci series: 0, 1, 1, 2, 3... etc. Kepler drew attention to the relationship of this sequence to the golden proportion: “It is arranged in such a way that the two lower terms of this never-ending proportion add up to the third term, and any two last terms, if added, give the next term, and the same proportion is maintained ad infinitum " Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden ratio in all its manifestations.
Leonardo da Vinci also devoted a lot of time to studying the features of the golden ratio; most likely, the term itself belongs to him. His drawings of a stereometric body formed by regular pentagons prove that each of the rectangles obtained by section gives the aspect ratio in the golden division.
Over time, the golden ratio rule became an academic routine, and only the philosopher Adolf Zeising gave it a second life in 1855. He brought the proportions of the golden section to the absolute, making them universal for all phenomena of the surrounding world. However, his “mathematical aesthetics” caused a lot of criticism.
Nature
Even without going into calculations, the golden ratio can be easily found in nature. So, the ratio of the tail and body of a lizard, the distances between the leaves on a branch fall under it, there is a golden ratio in the shape of an egg, if a conditional line is drawn through its widest part.
The Belarusian scientist Eduard Soroko, who studied the forms of golden divisions in nature, noted that everything growing and striving to take its place in space is endowed with the proportions of the golden section. In his opinion, one of the most interesting forms is spiral twisting.
Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Goethe later noted nature’s attraction to spiral forms, calling the spiral the “curve of life.” Modern scientists have found that such manifestations of spiral forms in nature as a snail shell, the arrangement of sunflower seeds, spider web patterns, the movement of a hurricane, the structure of DNA and even the structure of galaxies contain the Fibonacci series.
Human
Fashion designers and clothing designers make all calculations based on the proportions of the golden ratio. Man is a universal form for testing the laws of the golden ratio. Of course, by nature, not all people have ideal proportions, which creates certain difficulties with the selection of clothes.
In Leonardo da Vinci's diary there is a drawing of a naked man inscribed in a circle, in two superimposed positions. Based on the research of the Roman architect Vitruvius, Leonardo similarly tried to establish the proportions of the human body. Later, the French architect Le Corbusier, using Leonardo’s “Vitruvian Man,” created his own scale of “harmonic proportions,” which influenced the aesthetics of 20th-century architecture.
Adolf Zeising, studying the proportionality of a person, did a colossal job. He measured about two thousand human bodies, as well as many ancient statues, and concluded that the golden ratio expresses the average statistical law. In a person, almost all parts of the body are subordinate to it, but the main indicator of the golden ratio is the division of the body by the navel point.
As a result of measurements, the researcher found that the proportions of the male body 13:8 are closer to the golden ratio than the proportions of the female body - 8:5.
The art of spatial forms
The artist Vasily Surikov said “that in composition there is an immutable law, when in a picture you cannot remove or add anything, you cannot even add an extra point, this is real mathematics.” For a long time artists follow this law intuitively, but after Leonardo da Vinci, the process of creating a painting can no longer be accomplished without solving geometric problems. For example, Albrecht Durer used the proportional compass he invented to determine the points of the golden section.
Art critic F.V. Kovalev, having examined in detail Nikolai Ge’s painting “Alexander Sergeevich Pushkin in the village of Mikhailovskoye,” notes that every detail of the canvas, be it a fireplace, a bookcase, an armchair, or the poet himself, is strictly inscribed in golden proportions.
Researchers of the golden ratio tirelessly study and measure architectural masterpieces, claiming that they became such because they were created according to the golden canons: their list includes the Great Pyramids of Giza, Notre Dame Cathedral, St. Basil's Cathedral, and the Parthenon.
And today, in any art of spatial forms, they try to follow the proportions of the golden section, since, according to art critics, they facilitate the perception of the work and form an aesthetic feeling in the viewer.
Word, sound and film
The forms of temporary art in their own way demonstrate to us the principle of the golden division. Literary scholars, for example, have noticed that the most popular number of lines in poems of the late period of Pushkin’s work corresponds to the Fibonacci series - 5, 8, 13, 21, 34.
The rule of the golden section also applies in individual works of the Russian classic. So the climax" Queen of Spades"is a dramatic scene between Herman and the Countess, ending with the death of the latter. The story has 853 lines, and the climax occurs on line 535 (853:535 = 1.6) - this is the point of the golden ratio.
Soviet musicologist E.K. Rosenov notes the amazing accuracy of the ratios of the golden section in the strict and free forms of the works of Johann Sebastian Bach, which corresponds to the thoughtful, concentrated, technically verified style of the master. This is also true for the outstanding works of other composers, where the most striking or unexpected musical solution usually occurs at the golden ratio point
Film director Sergei Eisenstein deliberately coordinated the script of his film “Battleship Potemkin” with the rule of the golden ratio, dividing the film into five parts. In the first three sections the action takes place on the ship, and in the last two - in Odessa. The transition to scenes in the city is the golden middle of the film.

Let's find out what the ancient Egyptian pyramids, Leonardo da Vinci's Mona Lisa, a sunflower, a snail, a pine cone and human fingers have in common?

The answer to this question is hidden in the amazing numbers that have been discovered Italian medieval mathematician Leonardo of Pisa, better known by the name Fibonacci (born about 1170 - died after 1228), Italian mathematician . Traveling around the East, he became acquainted with the achievements of Arab mathematics; contributed to their transfer to the West.

After his discovery, these numbers began to be called after the famous mathematician. The amazing essence of the Fibonacci number sequence is that that each number in this sequence is obtained from the sum of the two previous numbers.

So, the numbers forming the sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, …

are called “Fibonacci numbers”, and the sequence itself is called the Fibonacci sequence.

There is one very interesting feature about Fibonacci numbers. When dividing any number from the sequence by the number in front of it in the series, the result will always be a value that fluctuates around the irrational value 1.61803398875... and sometimes exceeds it, sometimes does not reach it. (Approx. irrational number, i.e. a number whose decimal representation is infinite and non-periodic)

Moreover, after the 13th number in the sequence, this division result becomes constant until the infinity of the series... It was this constant number of divisions that was called the Divine proportion in the Middle Ages, and is now called the golden ratio, the golden mean, or the golden proportion. . In algebra, this number is denoted by the Greek letter phi (Ф)

So, Golden ratio = 1:1.618

233 / 144 = 1,618

377 / 233 = 1,618

610 / 377 = 1,618

987 / 610 = 1,618

1597 / 987 = 1,618

2584 / 1597 = 1,618

The human body and the golden ratio

Artists, scientists, fashion designers, designers make their calculations, drawings or sketches based on the ratio of the golden ratio. They use measurements from the human body, which was also created according to the principle of the golden ratio. Before creating their masterpieces, Leonardo Da Vinci and Le Corbusier took the parameters of the human body, created according to the law of the Golden Proportion.

The most main book For all modern architects, E. Neufert’s reference book “Building Design” contains basic calculations of the parameters of the human torso, which contain the golden proportion.

Proportions various parts our body is a number very close to the golden ratio. If these proportions coincide with the golden ratio formula, then the person’s appearance or body is considered ideally proportioned. The principle of calculating the gold measure on the human body can be depicted in the form of a diagram:

M/m=1.618

The first example of the golden ratio in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between a person’s foot and the navel point as a unit of measurement, then a person’s height is equivalent to the number 1.618.

In addition to this, there are several more basic golden proportions of our body:

* the distance from the fingertips to the wrist to the elbow is 1:1.618;

* the distance from shoulder level to the top of the head and the size of the head is 1:1.618;

* the distance from the navel point to the crown of the head and from shoulder level to the crown of the head is 1:1.618;

* the distance of the navel point to the knees and from the knees to the feet is 1:1.618;

* the distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1:1.618;

* the distance from the tip of the chin to the upper line of the eyebrows and from the upper line of the eyebrows to the crown is 1:1.618;

* the distance from the tip of the chin to the top line of the eyebrows and from the top line of the eyebrows to the crown is 1:1.618:

The golden ratio in human facial features as a criterion of perfect beauty.

In the structure of human facial features there are also many examples that are close in value to the golden ratio formula. However, do not immediately rush for a ruler to measure the faces of all people. Because exact correspondences to the golden ratio, according to scientists and artists, artists and sculptors, exist only in people with perfect beauty. Actually, the exact presence of the golden proportion in a person’s face is the ideal of beauty for the human gaze.

For example, if we sum up the width of the two front upper teeth and divide this sum by the height of the teeth, then, having obtained the golden ratio number, we can say that the structure of these teeth is ideal.

There are other embodiments of the golden ratio rule on the human face. Here are a few of these relationships:

*Face height/face width;

* Central point of connection of the lips to the base of the nose / length of the nose;

*Face height/distance from tip of chin to center point lip connections;

*Mouth width/nose width;

* Nose width / distance between nostrils;

* Distance between pupils / distance between eyebrows.

Human hand

It is enough just to bring your palm closer to you and look carefully at your index finger, and you will immediately find the formula of the golden ratio in it. Each finger of our hand consists of three phalanges.

* The sum of the first two phalanges of the finger in relation to the entire length of the finger gives the number of the golden ratio (with the exception of the thumb);

* In addition, the ratio between the middle finger and little finger is also equal to the golden ratio;

* A person has 2 hands, the fingers on each hand consist of 3 phalanges (except for the thumb). There are 5 fingers on each hand, that is, 10 in total, but with the exception of two two-phalanx thumbs, only 8 fingers are created according to the principle of the golden ratio. Whereas all these numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence:

The golden ratio in the structure of the human lungs

American physicist B.D. West and Dr. A.L. Goldberger, during physical and anatomical studies, established that the golden ratio also exists in the structure of the human lungs.

The peculiarity of the bronchi that make up the human lungs lies in their asymmetry. The bronchi consist of two main airways, one of which (the left) is longer and the other (the right) is shorter.

* It was found that this asymmetry continues in the branches of the bronchi, in all the smaller respiratory tracts. Moreover, the ratio of the lengths of short and long bronchi is also the golden ratio and is equal to 1:1.618.

Structure of the golden orthogonal quadrilateral and spiral

The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole.

In geometry, a rectangle with this aspect ratio came to be called the golden rectangle. Its long sides are in relation to its short sides in a ratio of 1.168:1.

The golden rectangle also has many amazing properties. The golden rectangle has many unusual properties. By cutting a square from the golden rectangle, the side of which is equal to the smaller side of the rectangle, we again obtain a golden rectangle of smaller dimensions. This process can be continued indefinitely. As we continue to cut off squares, we will end up with smaller and smaller golden rectangles. Moreover, they will be located in a logarithmic spiral, which is important in mathematical models of natural objects (for example, snail shells).

The pole of the spiral lies at the intersection of the diagonals of the initial rectangle and the first vertical one to be cut. Moreover, the diagonals of all subsequent decreasing golden rectangles lie on these diagonals. Of course, there is also the golden triangle.

English designer and esthetician William Charlton stated that people find spiral shapes pleasing to the eye and have been using them for thousands of years, explaining it this way:

“We like the look of a spiral because visually we can easily look at it.”

In nature

* The rule of the golden ratio, which underlies the structure of the spiral, is found in nature very often in creations of unparalleled beauty. The most illustrative examples— the spiral shape can be seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, the structure of rose petals, etc.;

* Botanists have found that in the arrangement of leaves on a branch, sunflower seeds or pine cones, the Fibonacci series is clearly manifested, and therefore the law of the golden ratio is manifested;

The Almighty Lord established a special measure for each of His creations and gave it proportionality, which is confirmed by examples found in nature. One can give a great many examples when the growth process of living organisms occurs in strict accordance with the shape of a logarithmic spiral.

All springs in the spiral have the same shape. Mathematicians have found that even with an increase in the size of the springs, the shape of the spiral remains unchanged. There is no other form in mathematics that has the same unique properties like a spiral.

The structure of sea shells

Scientists who studied the internal and external structure of shells soft-bodied molluscs, living at the bottom of the seas, stated:

“The inner surface of the shells is impeccably smooth, while the outer surface is completely covered with roughness and irregularities. The mollusk was in a shell and for this the inner surface of the shell had to be perfectly smooth. External corners-bends of the shell increase its strength, hardness and thus increase its strength. The perfection and amazing intelligence of the structure of the shell (snail) is amazing. The spiral idea of ​​shells is a perfect geometric form and is amazing in its honed beauty."

In most snails that have shells, the shell grows in the shape of a logarithmic spiral. However, there is no doubt that these unreasonable creatures not only have no idea about the logarithmic spiral, but do not even have the simplest mathematical knowledge to create a spiral-shaped shell for themselves.

But then how were these unreasonable creatures able to determine and choose for themselves the ideal form of growth and existence in the form of a spiral shell? Could these living creatures, which the scientific world calls primitive life forms, calculate that the logarithmic shell shape would be ideal for their existence?

Of course not, because such a plan cannot be realized without intelligence and knowledge. But neither primitive mollusks nor unconscious nature possess such intelligence, which, however, some scientists call the creator of life on earth (?!)

Trying to explain the origin of such even the most primitive form of life by a random combination of certain natural circumstances is absurd, to say the least. It is clear that this project is a conscious creation.

Biologist Sir D'arky Thompson calls this type of growth of sea shells "growth form of dwarves."

Sir Thompson makes this comment:

“There is no simpler system than the growth of sea shells, which grow and expand in proportion, maintaining the same shape. The most amazing thing is that the shell grows, but never changes shape.”

The Nautilus, measuring several centimeters in diameter, is the most striking example of the gnome growth habit. S. Morrison describes this process of nautilus growth as follows, which seems quite difficult to plan even with the human mind:

“Inside the nautilus shell there are many compartments-rooms with partitions made of mother-of-pearl, and the shell itself inside is a spiral expanding from the center. As the nautilus grows, another room grows in the front part of the shell, but this time it is larger than the previous one, and the partitions of the room left behind are covered with a layer of mother-of-pearl. Thus, the spiral expands proportionally all the time.”

Here are just some types of spiral shells with a logarithmic growth pattern in accordance with their scientific names:
Haliotis Parvus, Dolium Perdix, Murex, Fusus Antiquus, Scalari Pretiosa, Solarium Trochleare.

All discovered fossil remains of shells also had a developed spiral shape.

However, the logarithmic growth form is found in the animal world not only in mollusks. The horns of antelopes, wild goats, rams and other similar animals also develop in the form of a spiral according to the laws of the golden ratio.

Golden ratio in the human ear

In the human inner ear there is an organ called Cochlea (“Snail”), which performs the function of transmitting sound vibration. This bony structure is filled with fluid and is also shaped like a snail, containing a stable logarithmic spiral shape = 73º 43'.

Animal horns and tusks developing in a spiral shape

The tusks of elephants and extinct mammoths, the claws of lions and the beaks of parrots are logarithmic in shape and resemble the shape of an axis that tends to turn into a spiral. Spiders always weave their webs in the form of a logarithmic spiral. The structure of microorganisms such as plankton (species globigerinae, planorbis, vortex, terebra, turitellae and trochida) also have a spiral shape.

Golden ratio in the structure of microcosms

Geometric shapes are not limited to just a triangle, square, pentagon or hexagon. If we connect these figures in different ways with each other, we get new three-dimensional geometric figures. Examples of this are figures such as a cube or a pyramid. However, besides them, there are also other three-dimensional figures that we have not encountered in everyday life, and whose names we hear, perhaps for the first time. Among such three-dimensional figures are the tetrahedron (regular four-sided figure), octahedron, dodecahedron, icosahedron, etc. The dodecahedron consists of 13 pentagons, the icosahedron of 20 triangles. Mathematicians note that these figures are mathematically very easily transformed, and their transformation occurs in accordance with the formula of the logarithmic spiral of the golden ratio.

In the microcosm, three-dimensional logarithmic forms built according to golden proportions are ubiquitous . For example, many viruses have a three-dimensional geometric shape icosahedron. Perhaps the most famous of these viruses is the Adeno virus. The protein shell of the Adeno virus is formed from 252 units of protein cells arranged in a certain sequence. At each corner of the icosahedron there are 12 units of protein cells in the shape of a pentagonal prism and spike-like structures extend from these corners.

The golden ratio in the structure of viruses was first discovered in the 1950s. scientists from Birkbeck College London A. Klug and D. Kaspar. 13 The Polyo virus was the first to display a logarithmic form. The form of this virus turned out to be similar to the form of the Rhino 14 virus.

The question arises, how do viruses form such complex three-dimensional shapes, the structure of which contains the golden ratio, which are quite difficult to construct even with our human mind? The discoverer of these forms of viruses, virologist A. Klug, gives the following comment:

“Dr. Kaspar and I showed that for the spherical shell of the virus, the most optimal shape is symmetry such as the icosahedron shape. This order minimizes the number of connecting elements... Most of Buckminster Fuller's geodesic hemispherical cubes are built on a similar geometric principle. 14 Installation of such cubes requires an extremely accurate and detailed explanatory diagram. Whereas unconscious viruses themselves construct such a complex shell from elastic, flexible protein cellular units.”